Two-point correlation functions of scaling fields in the Dirac theory on the Poincaré disk

A result from Palmer, Beatty and Tracy suggests that the two-point function of certain spinless scaling fields in a free Dirac theory on the Poincaré disk can be described in terms of Painlevé VI transcendents. We complete and verify this description by fixing the integration constants in the Painlevé VI transcendent describing the two-point function, and by calculating directly in a Dirac theory on the Poincaré disk the long distance expansion of this two-point function and the relative normalization of its long and short distance asymptotics. The long distance expansion is obtained by developing the curved-space analogue of a form factor expansion, and the relative normalization is obtained by calculating the one-point function of the scaling fields in question. The long distance expansion in fact provides part of the solution to the connection problem associated with the Painlevé VI equation involved. Calculations are done using the formalism of angular quantization.